Summation Notation
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Summation Notation, sometimes called Sigma Notation, is a shorthand way of writing a long sum of numbers using the symbol , the Greek capital letter sigma.

Translating a simple sum into summation notation
Let's say we have a long string of integers being summed up:
There's a pattern to this sum: we start at 1, we go up to 20, and in between, each number we add to our total is one greater than the number that came before it. Because of this pattern, we should be able to describe the sum to someone else by explaining the pattern to them, instead of having to write the whole sum out. This is what summation notation does. In summation notation, the above sum looks like this:
Here's what the different pieces of the notation mean:
The sigma tells us that we're going to be adding up several things according to a pattern, and that our dummy variable will be increasing by 1 at each stage in the patern. 
The symbols to the right of the sigma tell us what the terms in our sum will look like. In this case, the i tells us that each piece of our sum is a single integer.  
The symbols under the sigma tell us what our dummy variable is, and what value it starts at. In this case, the i=1 tells us that our dummy variable is i, and that its starting value is 1. 
The symbols on top of the sigma tell us what value our dummy variable stops at. In this case, we're going to stop our sum when i is 20. 
So to get from the summation notation back to our original sum, first we take and replace it with a 1. Next, we take again, but this time we replace it with a 2, and add it to our previous 1. The next time, we replace with 3, and so on, until we've replaced with 20, at which point we stop.
As we can see in the example above, dummy variables are placeholders. We use them to make the outline of a pattern, but then we need to fill them in with something else.
We can think of dummy variables as being similar to cupcake liners. When we're making cupcakes, we need to set out the liners to see where our cupcakes will go. The liners create an outline and define the space where our cupcakes will be.
When baking, we don't actually have any cupcakes until we fill the liners with batter. Similarly, with summation notation, we don't actually have a sum until we replace the dummy variable with numbers.
In summation notation, we fill our dummy variable with a different number each time. In our cupcake analogy, we can imagine that we're making colored cupcakes: in each cupcake liner, we add one more drop of food coloring.

 This one was tricky  did you notice that the dummy variable was set to start at 2 and not 1?

Indexing
Let's look at another sum written with summation notation:
How do we translate this back into a long sum? Let's just ignore that for now, and follow the rules we used above. The first term in our sum is whatever symbols are to the right of the sigma, but with replaced by 1. The second term is the same, but with replaced by 2. In the third term, is replaced by 3, in the fourth term it's replaced by 4, and we stop at the fifth term, where is replaced by 5. Now we have something that looks like this:
But what are , , , , and ? Well, they could be unknowns. If
were part of a system of equations in five variables, we might write
instead to save space.
But , , , , and can also be part of an indexed list or sequence. If we have a sequence of numbers, such as , we can use indexing to talk about the numbers in the sequence. So can mean "the first item the sequence", which happens to be the number 0. Similarly, , because 6 is the second number in the sequence; and , because 14 is the third number in the sequence. If our sum is talking about the sequence , then it is asking us to add up the first five terms in that sequence:
Sums to n and ∞
Summation and distribution
Matrices and doubleindexing
Polynomials
Common power sums